Riesz representation theorem

For the theorems relating linear functionals to measures, see Riesz–Markov–Kakutani representation theorem.

In functional analysis the Riesz representation theorem describes the dual of a Hilbert space. It is named in honour of Frigyes Riesz.

Statement

If $H$ a Hilbert Space then $H'=B(H,\mathbb {F} )=\{\{({\underline {y}},\langle {\underline {y}},{\underline {x}}\rangle ):{\underline {y}}\in H\}:{\underline {x}}\in H\}$

Proof

"$\supseteq$": linearity comes from the fact that the inner product, by definition, is linear in the first argument and boundedness comes from the Cauchy-Schwartz inequality.

Now to deal with "$\subseteq$": Let ${\underline {\phi }}\in H'$; if ${\underline {\phi }}={\underline {0}}$ then ${\underline {\phi }}=\{({\underline {y}},\langle {\underline {y}},{\underline {0}}\rangle ):{\underline {y}}\in H\}$.

Now suppose ${\underline {\phi }}\neq {\underline {0}}$ and let ${\underline {x}}\in H$; ${\overline {ker({\underline {\phi }})}}=ker({\underline {\phi }})$ and $ker({\underline {\phi }})\leqslant H$ so by the Hilbert projection theorem $H=ker({\underline {\phi }})\oplus (ker({\underline {\phi }}))^{\perp }$.

Well, $ker(\phi )\neq H$ so $(ker({\underline {\phi }}))^{\perp }\neq \{{\underline {0}}\}$ so let ${\underline {z}}\in \{{\underline {z}}\in (ker({\underline {\phi }}))^{\perp }:\|{\underline {z}}\|_{H}=1\}$

Then by the linearity of ${\underline {\phi }}$, ${\underline {z}}{\underline {\phi }}({\underline {x}})-{\underline {x}}{\underline {\phi }}({\underline {z}})\in ker({\underline {\phi }})$, and so $\langle {\underline {z}}{\underline {\phi }}({\underline {x}})-{\underline {x}}{\underline {\phi }}({\underline {z}}),{\underline {z}}\rangle =0$ and so ${\underline {\phi }}({\underline {x}})\|{\underline {z}}\|_{H}=\langle {\underline {x}}{\underline {\phi }}({\underline {z}}),{\underline {z}}\rangle$ so ${\underline {\phi }}({\underline {x}})=\langle {\underline {x}},{\underline {z}}{\overline {{\underline {\phi }}({\underline {z}})}}\rangle$ as $\|{\underline {z}}\|_{H}=1$.

So ${\underline {\phi }}=\{({\underline {y}},\langle {\underline {y}},{\underline {z}}{\overline {{\underline {\phi }}({\underline {z}})}}\rangle ):{\underline {y}}\in H\}\in \{\{({\underline {y}},\langle {\underline {y}},{\underline {x}}\rangle ):{\underline {y}}\in H\}:{\underline {x}}\in H\}$

The Hilbert space representation theorem

This theorem establishes an important connection between a Hilbert space and its (continuous) dual space: if the underlying field is the real numbers, the two are isometrically isomorphic; if the field is the complex numbers, the two are isometrically anti-isomorphic. The (anti-) isomorphism is a particular natural one as will be described next.

Let H be a Hilbert space, and let H* denote its dual space, consisting of all continuous linear functionals from H into the field R or C. If x is an element of H, then the function φ_x, defined by

$\phi _{x}(y)=\left\langle y,x\right\rangle \quad \forall y\in H$

where $\langle \cdot ,\cdot \rangle$ denotes the inner product of the Hilbert space, is an element of H*. The Riesz representation theorem states that every element of H* can be written uniquely in this form.

Theorem. The mapping Φ: H → H* defined by Φ(x) = φ_x is an isometric (anti-) isomorphism, meaning that:

• Φ is bijective.
• The norms of x and Φ(x) agree: $\Vert x\Vert =\Vert \Phi (x)\Vert$.
• Φ is additive: $\Phi (x_{1}+x_{2})=\Phi (x_{1})+\Phi (x_{2})$.
• If the base field is R, then $\Phi (\lambda x)=\lambda \Phi (x)$ for all real numbers λ.
• If the base field is C, then $\Phi (\lambda x)={\bar {\lambda }}\Phi (x)$ for all complex numbers λ, where ${\bar {\lambda }}$ denotes the complex conjugation of λ.

The inverse map of Φ can be described as follows. Given an element φ of H*, the orthogonal complement of the kernel of φ is a one-dimensional subspace of H. Take a non-zero element z in that subspace, and set $x={\overline {\varphi (z)}}\cdot z/{\left\Vert z\right\Vert }^{2}$. Then Φ(x) = φ.

Historically, the theorem is often attributed simultaneously to Riesz and Fréchet in 1907 (see references).

In the mathematical treatment of quantum mechanics, the theorem can be seen as a justification for the popular bra-ket notation. When the theorem holds, every ket $|\psi \rangle$ has a corresponding bra $\langle \psi |$, and the correspondence is unambiguous.

References

• M. Fréchet (1907). Sur les ensembles de fonctions et les opérations linéaires. C. R. Acad. Sci. Paris 144, 1414–1416.
• F. Riesz (1907). Sur une espèce de géométrie analytique des systèmes de fonctions sommables. C. R. Acad. Sci. Paris 144, 1409–1411.
• F. Riesz (1909). Sur les opérations fonctionnelles linéaires. C. R. Acad. Sci. Paris 149, 974–977.
• J. D. Gray, The shaping of the Riesz representation theorem: A chapter in the history of analysis, Archive for History in the Exact Sciences, Vol 31(2) 1984–85, 127–187.
• P. Halmos Measure Theory, D. van Nostrand and Co., 1950.
• P. Halmos, A Hilbert Space Problem Book, Springer, New York 1982 (problem 3 contains version for vector spaces with coordinate systems).
• D. G. Hartig, The Riesz representation theorem revisited, American Mathematical Monthly, 90(4), 277–280 (A category theoretic presentation as natural transformation).
• "Riesz representation theorem", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Walter Rudin, Real and Complex Analysis, McGraw-Hill, 1966, ISBN 0-07-100276-6.
• Weisstein, Eric W. "Riesz Representation Theorem". MathWorld.
• "Proof of Riesz representation theorem for separable Hilbert spaces". PlanetMath.
• Proof of Riesz representation theorem in Hilbert spaces on Bourbawiki